Two-Relay Distributed Switch and Stay Combining

Transcription

1 1790 IEEE RANSACIONS ON COMMUNICAIONS VOL. 56 NO. 11 NOVEMBER 008 wo-relay Distributed Switch and Stay Combining Diomidis S. Michalopoulos Student Member IEEE and George K. Karagiannidis Senior Member IEEE Abstract We study a distributed version of switch-and-stay combining DSSC for systems that utilize two relays. In particular four different scenarios are considered depending on a whether or not the source-destination channel is taken into account and b the type of relaying i.e. decode and forward or amplify and forward. A performance analysis in terms of outage and bit error probability is presented when operating over Rayleigh fading channels. Numerical results demonstrate that two-relay DSSC achieves the same diversity gain and outage performance as if the best relay is selected for each transmission slot albeit simpler. Index erms Distributed switch and stay combining relaying channel. Fig. 1. he proposed setup. I. INRODUCION RELAYING transmissions have been recently proposed as a mean of attaining spatial diversity without using multiple antennas at either the transmitter or the receiver. Considering thus that relaying transmissions serve as a substitute of the common diversity techniques which have been extensively analyzed in the literature it naturally follows that they can be studied and thereby designed under that perspective. his alternative diversity-achieving concept was initially studied in [1] where a set of relaying protocols for the singlerelay scenario were proposed. he authors of [1] showed that utilizing channel knowledge can improve the performance by activating the relay and thus using only half of the degrees of freedom of the channel only when necessary. In [] the authors extended this single-relay diversity concept and proposed a distributed version of the well-known switchand-stay combining SSC technique where the same branch remains active as long as the signal-to-noise-ratio SNR of that branch is above a given threshold [3]- [4]. In cases where more than one relays are available opportunistic relaying [5] was shown to attain diversity gain on the order of the number of relays by activating only the best relay for each transmission slot representing thus a distributed version of selection combining [6 ch. 9.8]. In this letter we extend the distributed SSC DSSC concept proposed in [] for the case where two relaying terminals are utilized; this may be the case in practical scenarios where deep shadowing renders it difficult to achieve diversity with Paper approved by M. Chiani the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received February ; revised April and August his work was conducted within the framework of the Reinforcement Program of Human Research Manpower PENED 03 partially funded by the E.U.-European Social Fund 75% and the Greek Ministry of Development 5%. he authors are with the Wireless Communications Systems Group WCSG Electrical and Computer Engineering Department Aristotle University of hessaloniki 5414 hessaloniki Greece {dmixalo; geokarag}@auth.gr. Digital Object Identifier /COMM /08$5.00 c 008 IEEE single-relay usage. In particular we study a two-relay cooperative scheme where only a single relay is activated in each transmission slot in a fashion similar to SSC i.e. the same relay remains active as long as the corresponding equivalent SNR is sufficiently high. We consider four different scenarios depending on a whether the source-destination channel is taken into account together with the relaying ones or not and b the type of relaying i.e. decode and forward DF or amplify and forward AF. Closed-form expressions for the outage probability of the proposed schemes for each of the above cases are provided. Moreover the bit error probability BEP for the case of uncoded BPSK modulation is studied allowing for a broader view of the performance of the tworelay DSSC under different coding assumptions. II. SYSEM MODEL he system under consideration is depicted in Fig. 1. In particular we consider a source node S which wants to communicate with a destination one D. worelays namely R 1 and R are willing to assist this communication either by demodulating the received signal and then remodulating and forwarding it to D or by acting as simple analog repeaters i.e. amplifying and forwarding the signal to D without any further process. he former type of relaying is widely known as DF; the latter as AF. he relays are assumed to operate in the half-duplex mode; that is they cannot receive and trannsmit simultaneously but on different timeslots. Hence in the first subslot of each transmission slot they listen to the source whereas in the second subslot they send the processed data along with a packet that contains source-relay channel state information CSI to the destination. We denote by R i the relaying channel associated with R i with corresponding equivalent SNR represented by i i {1 }. In the sequel we use the subscript i to refer to both subscripts 1 and i.e. i {1 } and these relaying channels are termed branches since they actually represent the input branches of the virtual SSC. he active branch is denoted

2 MICHALOPOULOS and KARAGIANNIDIS: WO-RELAY DISRIBUED SWICH AND SAY COMBINING 1791 by A so that the event that the R i branch is active can be concisely written as A = R i. he instantaneous SNRs of the S-D S-R i and R i -D channels are represented by SRi and RiD respectively. Further we assume that these channels experience independent flat and slow Rayleigh fading with average SNRs denoted by SRi and RiD respectively. Also the transmission slots are considered small enough so that constant fading conditions during two consecutive slots can be assumed. A. Mode of Operation Being a distributed version of the SSC techniques the proposed system activates only one of the two relays during each transmission slot in a switch-and-stay fashion. More specifically in each transmission slot the destination compares the equivalent SNR of the active branch with a switching threshold denoted by. If the SNR is lower than thena branch-switching occurs. his is implemented by appropriate feedback sent to both relays which in the next slot switch from the active to the idle mode and vice versa. In a word the destination keeps receiving from and keeps estimating the equivalent SNR of a single branch regardless of the channel conditions of the other until the equivalent SNR of that branch falls below. Depending on the hardware complexity the destination can tolerate it can be adjusted to receive only from the relays or from both the source and the relays during the first and second subslot respectively. In the latter case the destination combines the received signals in a maximal ratio combiner MRC. Apparently employing a MRC at the destination enhances the performance; this however comes at the cost of complexity and in cases where the S-D channel is deeply shadowed this MRC employment might be useless. We note that the reader should not confuse this actual diversity combiner with the virtual SSC that our system employs as described above. III. WO-RELAY DSSC WIHOU MRC A HE DESINAION In cases where employing a diversity combiner at the destination is either unfeasible due to hardware constraints or just superfluous due to deep shadowing in the S-D channel the proposed system can be thought of as a virtual SSC scheme where the two input branches are R 1 and R. A. Relays Operate in the DF mode If DF relays are used the signal that reaches the destination through R i undergoes two demodulations in cascade. hus i is not trivially derived. In the sequel we adopt the outagebased definition for i ; specifically i is defined such that its cumulative density function CDF evaluated at the outage threshold SNR th where th = r 1 with r representing the target rate coincides with the outage probability of the R i branch i.e. F i th =Pr{O A = R i } where F Z stands for the CDF of the random variable Z and O denotes the outage event. Considering that Pr {O A = R i } is the probability of the union of the outage events corresponding to the S-R i and R i -D channels i is defined as i := min SRi RiD. 1 We emphasize here however that i is the quantity associated with the R i branch that is compared with and does not generally represent the equivalent SNR of that branch with the strict sense i.e. if BPSK modulation is used Pr {E A = R i i } 1 erfc i where E represents the bit-error event and erfc is the complementary error function. 1 Outage Performance: he outage probability of the proposed system is straightforwardly derived by utilizing the outage analysis of SSC systems [6 eq. 9.37] as F 1 F F 1 th F th F 1 F th < F P out th = 1 F F 1 th F th F 1 F th F 1 thf F th F 1 F 1 F where F i x = 1 exp x/ SRi exp x/rid. We note that the choice of significantly affects the outage performance: For a given th the outage probability is minimized by setting = th see [6 ch ] since in that case yields P out th = F 1 th F th 3 [ = 1 exp th exp ] th SRi RiD i.e. the optimal outage probability of DSSC equals that of a system that selects the best of the R 1 and R channels for each transmission slot. BEP Analysis: Let us denote with p Ri the steady-state selection probability of R i i.e.p Ri =Pr{A = R i } which has been evaluated in [4] as p Ri = = F j k=1 F k 1 exp exp SRj [ k=1 1 exp exp SRk Rj D Rk D ] 4 where j is the complement of i with respect to {1 } i.e. j =if i =1and vice versa. hen considering the system s mode of operation described in Section II-A the BEP can be expressed as Pr {E} = p Ri [F i Pr{E A = R j } 5 1 F i Pr {E A = R i and i }]. Assuming uncoded BPSK modulation the conditional BEP conditioned on the SNR is defined as Pr {E } = 1/ erfc. Moreover the Ri branch leads to an error if an error on either the S-R i or the R i -D link but not on both occurs. herefore the conditional BEP conditioned on the event A = R i and i is obtained by averaging over

3 179 IEEE RANSACIONS ON COMMUNICAIONS VOL. 56 NO. 11 NOVEMBER 008 the exponential probability density functions PDFs of SRi and RiD as Pr {E A = R i i } 1 1 = I I 1 SRi SRi RiD RiD I 1 I 1 6 SRi RiD SRi RiD where the auxiliary function I α β ω is defined as see [ Appendix] I α β ω = e αx erfc βx dx 7 ω = 1 β α eαω erfc βω α α β erfc α β ω. Consequently a closed-form expression for the BEP is derived by inserting 4 and 6 in 5. Note that the probabilities Pr {E A = R i } can be also expressed as shown in 6 by setting =0. B. Relays Operate in the AF mode Let us now consider that both relays operate in the AF mode. We further assume that the relays are capable of eliminating half of the propagated noise power by using the signalrotation technique proposed in [7] resulting in an equivalent SNR i = SRi RiD/ SRi RiD 1/. However in the sequel we focus on the following tight upper bound of i i = SR i RiD 8 SRi RiD which in fact corresponds to an ideal relay gain capable of inverting the attenuation in the S-R i link ignoring the noise. he same study regarding the bound of i was also conducted in [8]- [9] where the authors showed that 8 results in a tight bound of the corresponding performance metrics which is even tighter when the noise-reduction technique [7] is being used especially for medium and high SNRs. Note that contrary to the DF case i represents the metric of the R i branch that is compared with and is also the SNR associated with the performance corresponding to this branch. 1 Outage Performance: Similarly to the DF case the outage probability for the AF relaying scenario is obtained directly from and from 3 as well assuming that the optimal = th has been set by substituting F i with see [8] F i x =1 x e σ i x x i K 1 9 i i where σ i = SRi RiD = SRi RiD and K v stands for the modified Bessel function of the second kind and order v. BEP Analysis: For uncoded BPSK modulation an analytical expression for the BEP is obtained from 5 by averaging the conditional BEP expressions over the PDF of i given in [8 eq. 1]. Unfortunately such expression is not easily tractable and cannot be further simplified. In the high- SNR regime however the BEP for BPSK modulation can be approximated in closed-form by utilizing and the alternative definition of the conditional BEP Pr {E } = Q where Q is the Gaussian Q-function as follows: Let X be a Gaussian distributed random variable with zero-mean and unitary variance i.e. X N 0 1. Denoting with A the system SNR i.e. A = i if A = R i and using the definition of the Gaussian Q-function the BEP is expressed as = Pr {E} =Pr {X > } A =Pr 0 P out X e X dx = π 0 { A < X } 10 P out y e y π dy. y Using the approximation K 1 z 1/z for z<<1 [10 eq ] 9 yields F i x 1 e σ i x =1 exp x/ SRi exp x/rid. 11 Notice that 11 gives a high-snr approximation for F i which is identical with the CDF of min SRi RiD; thatis in the high-snr region we may define i as in 1 instead of 8 a fact which was also addressed in [11 Property 1]. By substituting 11 into and then inserting in 10 after some manipulations we infer Pr {E} 1 erf σ i 1e 1erfc σ 1 σ e 1 e σ σi i σ j e e j σ i i σ i where we have used the fact that π/zerf zx [1 eq ]. σ j 1e j σ σi i σ 1 j e e j x 1/ e zx dx = IV. WO-RELAY DSSC WIH MRC A HE DESINAION Now let us assume that the destination is equipped with a MRC so that it can optimally combine the signals received from the S-D and one of the R 1 R branches. Specifically in the first subslot of each transmission slot the destination receives the signal incident from S and inserts it into a timediversity MRC. At the same time the relays also receive the same signal but only the active relay as this is determined by the switch-and-stay process forwards it to the destination in the second subslot in order to be inserted into the MRC. hen the destination compares the SNR at the MRC output with the switching threshold and if this SNR is lower than it sends appropriate feedback to the relays indicating their next-slot transition from the active to the idle mode and vice versa. A. Relays Operate in the DF mode 1 Outage Performance: Using the outage-based definition of i i.e. i : F i th =Pr{O A = R i } together with the fact that an outage occurs if neither the direct nor the

4 MICHALOPOULOS and KARAGIANNIDIS: WO-RELAY DISRIBUED SWICH AND SAY COMBINING 1793 Fig.. he proposed scheme s optimal outage probability. Fig. 3. BEP performance of the proposed and the OR scheme assuming BPSK modulation and that no diversity combiner is employed at the destination. relayed branch together with the direct one can support the target rate r it holds 1 F i th = F SRi th F th 13 1 F SRi th F gi th where g i = i is the SNR at the combiner output when the R i branch is active; F gi is thus derived by convoluting F Ri D with the exponential PDF of. Hence 13 yields F i th = 1 e th SRi 1 e th 14 e th RiD SRi 1 e th Ri D 1 e th RiD and therefore the outage probability is derived by substituting 14 in or in 3 in case of setting = th. B. Relays Operate in the AF mode In such case i is expressed as i = SR i RiD. 15 SRi RiD 1 Outage Performance: An analytical expression for the CDF of i is obtained by convoluting 9 with the exponential PDF of f ; however such expression is not easily tractable and cannot yield a closed-form expression for the F i. In the high SNR regime using the approximation [10 eq ] we infer 1 e σ i x i σ i 1 e x F i x. 16 σ i BEP Analysis: Working similarly as in Section III-B we may obtain a high-snr approximation for the BEP by utilizing and the indefinite integral x 1/ e zx dx = π/zerf zx; this BEP expression is given in eq. 17 shown at the top of the next page. 1 We note that the protocol presented here differs from the DF protocol proposed in [1] in the sense that even if the source-relay link is in outage the destination still attempts to decode the message from the source-destination channel. V. NUMERICAL EXAMPLES AND DISCUSSION In this section we present a schematic illustration of the proposed schemes performance along with that of the opportunistic relaying OR where in each transmission slot the branch with the highest instantaneous SNR is selected i.e. A = R arg maxi {1} i [5].Inallfigures the S-R i and R i -D channels are assumed to experience independent and identically distributed Rayleigh fading with average value equal to four times that of the also Rayleigh distributed S-D channel i.e. SRi = RiD =4. Fig. depicts the outage probability of the proposed scheme versus the normalized value of SRi with respect to th. he switching threshold used is the optimal one i.e. = th and thus the curves shown in this figure portray also the outage probability of the corresponding OR schemes. In Fig. both complexity-tolerance assumptions i.e. with and without MRC at D and both relaying modes DF and AF are considered. We note that for the AF case with MRC at the destination the solid line was derived through simulations; the dotted one using 16 and 3. Interestingly one may notice that for the former assumption i.e. no diversity combiner employed at the destination the DF performance is silghtly better than the AF one whereas for the latter AF outperforms DF. his is due to the fact that the combining weights employed by the MRC in the DF case do not take into account the source-relay channel resulting in sub-optimum combining of the received signals; this fact was also addressed in [11]. he BEP performance of the DSSC and the OR systems versus SRi is depicted in Figs. 3 and 4 assuming uncoded BPSK modulation; in the former figure the destination is assumed not to employ any diversity combiner whereas in the latter it employs a MRC. Each curve in these figures was generated by using the optimal switching threshold which is derived numerically by minimizing the corresponding BEP expressions with respect to. For the case of DF relaying with MRC at the destination the curves are derived via simulations using the same switching threshold as that with the no MRC case. As expected all OR schemes outperform the equivalent DSSC ones albeit achieving the same diversity order since generally speaking selection combining can be seen as an

5 1794 IEEE RANSACIONS ON COMMUNICAIONS VOL. 56 NO. 11 NOVEMBER 008 Pr {E} [ σ 1σ 3/ erf 1 1 σ j σ i σ j erf 1 σ i [ 1 ] e σ i σ i 1 e j σ j 1 e σij e σ i j 1 e σ j 3/ i σ i erfc [ 1 e σ i σ i j σ j e σ j j 1 σ j i σ jerf σ i [ 1 e σ i σ i 1 e ] 1 1 e σ i ] erfc ] 17 1 e j σ j 1 e i 1e σ i σ i 1e 1 erfc σ i 1 1 [ 1 ] e σ i σ i 1 e j σ j σ i j 1 e σ j j σ j 1 e i 1 1e σ i σ i σ i 1e [ 1 ] e σ i σ i 1 e j σ j period offering thus a simpler alternative of relay selection while still attaining the same diversity gain as well as identical outage probability. ACKNOWLEDGMEN he authors would like to thank Mr. Athanasios Lioumpas for helpful discussions as well as an anonymous reviewer for his/her insightful comments and suggestions. Fig. 4. BEP performance of the proposed and the OR scheme assuming BPSK modulation and MRC at the destination. optimal yet more complex implementation of SSC. he simplicity of DSSC lies in the fact that only a single branch is estimated in each transmission slot and that feedback to the relays is not sent continuously but only after a branch-switching decision. hat is neither global CSI nor feedback in each transmission slot is needed. Finally it is interesting to note the difference between the slope of the BEP curves for the DF case without MRC employment and the corresponding outage curves shown in Fig. leading to superior AF performance compared to the DF one. his stems from the fact that contrarily to Fig. uncoded modulation is assumed in Figs. 3 and 4 resulting in a significant DF performance degradation due to error propagations. VI. CONCLUSIONS We presented the concept of two-relay distributed switch and stay combining where only one out of two available relays is activated in a switch and stay fashion. his allows for only a single end-to-end branch to be estimated in each transmission REFERENCES [1] J. N. Laneman D. N. C. se and G. W. Wornell Cooperative diversity in wireless networks: Efficient protocols and outage behavior IEEE rans. Inform. heory vol. 50 pp Dec [] D. S. Michalopoulos and G. K. Karagiannidis Distributed switch and stay combining DSSC with a single decode and forward relay IEEE Communications Letters vol. 11 May 007. [3] A. A. Abu-Dayya and N. C. Beaulieu Analysis of switched diversity systems on generalized-fading channels IEEE rans. Commun. vol. 4 pp Nov [4] C. ellambura A. Annamalai and V. K. Bhargava Unified analysis of switched diversity systems in independent and correlated fading channels IEEE rans. Commun. vol. 49 pp Nov [5] A. Bletsas A. Khisti D. P. Reed and A. Lippman A simple cooperative diversity method based on network path selection IEEE J. Selec. Areas Commun. vol. 4 pp Mar [6] M. K. Simon and M.-S. Alouini Digital Communication over Fading Channels nd ed. New York: Wiley 005. [7] N. C. Beaulieu and J. Hu A noise reduction amplify and forward relay protocol for distributed spatial diversity IEEE Commun. Let. vol. 10 pp Nov [8] P. A. Anghel and M. Kaveh Exact symbol error probability of a cooperative network in a Rayleigh-fading environment IEEE rans. Wireless Commun. vol. 3 pp Sep [9] A. Ribeiro X. Cai and G. Giannakis Symbol error probabilities for general cooperative links IEEE rans. Wireless Commun. vol. 4 pp May 005. [10] M. Abramovitz and I. A. Stegun Handbook of Mathematical Functions with Formulas Graphs and Mathematical ables 9thed. NewYork: Dover 197. [11]. Wang A. Cano G. B. Giannakis and J. N. Laneman Highperformance cooperative demodulation with decode-and-forward relays IEEE rans. Commun. vol. 55 pp Jul [1] I. S. Gradshteyn and I. M. Ryzhik able of Integrals Series and Products 6th ed. New York: Academic 000.